A124527 -id:A124527 - cp's OEIS frontend

A124526 Triangle, read by rows, where T(n,k) = A049020([n/2],k)*A049020([(n+1)/2],k).

1, 1, 1, 1, 2, 3, 4, 9, 1, 10, 30, 6, 25, 100, 36, 1, 75, 370, 186, 10, 225, 1369, 961, 100, 1, 780, 5587, 4960, 750, 15, 2704, 22801, 25600, 5625, 225, 1, 10556, 101774, 136960, 39000, 2325, 21, 41209, 454276, 732736, 270400, 24025, 441, 1, 178031, 2199262, 4110512, 1849120, 217000, 6027, 28, 769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1, 3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36

Offset: 0

Paul D. Hanna, Nov 08 2006

Row n has 1+floor(n/2) terms.
T(n,0) = A124419(n).
A124418(n,k) = k!*T(n,k) (conjecture).
A000110(n) = Sum_{k=0..[n/2]} k!*T(n,k), where A000110 is the Bell numbers.
Inspired by triangle A124418 and the work of Emeric Deutsch.

			Triangle begins:
1;
1;
1, 1;
2, 3;
4, 9, 1;
10, 30, 6;
25, 100, 36, 1;
75, 370, 186, 10;
225, 1369, 961, 100, 1;
780, 5587, 4960, 750, 15;
2704, 22801, 25600, 5625, 225, 1;
10556, 101774, 136960, 39000, 2325, 21;
41209, 454276, 732736, 270400, 24025, 441, 1;
178031, 2199262, 4110512, 1849120, 217000, 6027, 28;
769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1;
3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36; ...

Cf. A124418, A124419, A049020; A124527, A124528, A124529.

S[n_, k_] = Sum[StirlingS2[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := S[Floor[n/2], k] S[Floor[(n+1)/2], k];
Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Nov 02 2020 *)

{T(n,k) = (n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+1)\2),k)}
for(n=0,15, for(k=0,n\2, print1(T(n,k),", "));print(""))

T(n,k) = A049020([n/2],k) * A049020([(n+1)/2],k), where A049020(n,k) = Sum_{i=0..n} S2(n,i) * C(i,k) and S2(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind).

A124528 a(n) = Sum_{k=0..n} 2^k*A124526(n,k) for n>=0.

Original entry on oeis.org

1, 1, 3, 8, 26, 94, 377, 1639, 7623, 38034, 199338, 1111816, 6442481, 39478219, 249507483, 1659172454, 11321526022, 80944313164, 591617080305, 4514822914133, 35120998653271, 284407875530728, 2342407874087454

Offset: 0

Paul D. Hanna, Nov 08 2006

nonn

Cf. A124526, A124527, A124529.

{a(n)=sum(k=0,n\2,2^k*(n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+1)\2),k))}

A124529 a(n) = Sum_{k=0..n} k!*A124526(n+k,k) for n>=0.

Original entry on oeis.org

1, 2, 6, 29, 190, 1562, 15457, 179034, 2377092, 35599701, 593731310, 10914169312, 219252994039, 4779086510108, 112341582757512, 2833025331800643, 76293601822430388, 2185288262904326236, 66338823231846583471

Offset: 0

Paul D. Hanna, Nov 08 2006

nonn

Equals diagonal sums of triangle A124418: a(n) = Sum_{k=0..n} A124418(n+k,k) for n>=0 (conjecture).

Cf. A124526, A124527, A124528; A124418.

{a(n)=sum(k=0,n,k!*((n+k)\2)!*((n+k+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+k)\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+k+1)\2),k))}

cp's OEIS Frontend

A124526 Triangle, read by rows, where T(n,k) = A049020([n/2],k)*A049020([(n+1)/2],k).

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A124528 a(n) = Sum_{k=0..n} 2^k*A124526(n,k) for n>=0.

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A124529 a(n) = Sum_{k=0..n} k!*A124526(n+k,k) for n>=0.

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